Certainp-regularH-spaces

Higher homotopy associativity of power maps on finite H-spaces

“…We prove by induction on n. If n = 1, then the result is proved by Hubbuck-Mimura [11] and Iwase [13,Proposition 0.7]. Assume that the result is true for n − 1.…”

“…which is a contradiction, and so we have (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14). Then for dimensional reasons and by (3)(4)(5)(6)(7)(8)(9)(10)(11)(12),…”

“…Then from the same arguments as above, we have that A * ( X) is a D 2 -algebra. By Kanemoto [15,Lemma 3], there is a generator y k ∈ QA 2(p−1) ( X) for some 1 ≤ k < l. Let K be the ideal of A * ( X) generated by y i with i = k. From the same reason as (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14), we see that P 1 (y i ) ∈ K for some 1 ≤ i ≤ l. Then for dimensional reasons, we see that P 1 (y i ) contains the term by 2 k with b = 0 in Z/p. By Proposition 3.1, we have a contradiction, and so X does not admit an AC 2 -form.…”

“…which implies that P 1 (y i ) contains the term ay n l with a = 0 in Z/p by (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14). By Proposition 3.1, we have y l ∈ P 1 QA 2(m l −p+1) ( X), which causes a contradiction since m l < p.…”

Truncated polynomial algebras over the Steenrod algebra

“…. , x2nj p+1 [4] . If an addition, 1 Comparing this list with the list of possible dimensions of the generators of a polynomial algebra B = Zp[y2n, y2 ,n] over the Steenrod algebra [3], we see that the only pairs which do not appear in the latter are {6,10} and {8, 20} when p = 11 and {6,10}, {8,16} and {12,18} when p = 13.…”